Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?

I'm Eight

Stage: 1, 2, 3 and 4 Challenge Level:

I'm Eight

When I went into a classroom earlier this week a child rushed up to tell me she was 8 that day!

Well, Happy Birthday to everyone who has a birthday today!

If you are 8 then this could be for you, but if it is another number then you just change the 8 to whatever your age is today.

There is not a lot to say to introduce this challenge. It's really just to find a great variety of ways of asking questions which make $8$.

Things like $6 + 2$, $22 - 14$, etc.

But you need to get examples that use all the different mathematical ideas that you know about.

$1$) So you could show some multiplications and some divisions.

$2$) If you know about fractions then you can add or subtract numbers involving fractions. You could also ask questions like "What is half of $16$?''; "What is four-fifths of 10?'' and so on.

$3$) If you've come across decimals then do a few of those also, perhaps using all the four rules [addition, subtraction, multiplication and division].

And so on.

Use whatever mathematics you know to find as many different ways of getting the answer $8$.

You may find some patterns that would go on for ever and ever. If you do, just put down a few, and then see if you can describe how the pattern works.

So if you're $8$ years old maybe you'll write something like this:

$16 \div 2$, $8 \div 1$, $4 + 4$, $2 + 6$, $9 - 1$, $12 - 4$

$1 + 1 + 1 + 1 + 1 + 1 + 1 + 1, 2 + 2 + 2 + 2$

$15 - 3 - 2 - 1 - 1, 5 + 3 + 6 - 3 - 3$

and so on.

But if you're much much older you may write something like:-

$4 \sin (\pi/2) + \sqrt{5^2 - 3^2}$

Whatever your age, and whatever ones you get caught up with, have a look at the ways that you can make new ones that have a similar pattern.

Your "What would happen if ...?'' questions may be a little different from our usual ones.

The 8 year old might ask "I wonder what would happen if I tried to use multiplication and addition to make 8?''

The much older person (17 years old perhaps) may well ask "I wonder what would happen if I used matrices?''

Why do this problem?

As a number activity I have found this to be one of the very best for both engaging pupils in thoughtful work and for getting them to push forward their own understanding of number.

Possible approach

Working at the front on some board to display the pupils' ideas I have started by vaguely grouping their suggestions into the separate four rules of number: addition, subtraction, multiplication and division. After a while I asked the youngsters if there are any more, but just wrote 'etc' to indicate that there were some more if they thought so.

Then I pointed out that they had used two numbers for each suggestion; so, could they use three or four numbers and, by adding, get to 8? After more examples I asked if they could start with a biggish number and then take some away and then have to take some more away in order to end up with only 8?

If and when slip-ups occur (suppose they have suggested 15 - 6 - 6), I would ask what has to happen to the answer so far, so that the answer can get to 8. I then talked with the children about the fact they they can use any mathematics that they understand as long as the answer is 8.

Key questions

(Not assuming that you follow their recording - which may have to be a bit unorthodox - it's good to pose questions that help you to know how the child was thinking.)

Tell me what you are doing here.

How have you got these?

Could you find any more like that?

Possible extension

1) You might find that, for example, a pupil continues with loads and loads of subtractions raising the starting number by just one each time. My own feeling is that I'll allow that to happen for the first two or three lessons in which I use this starting point. If they carry on in the next lesson I would encourage them to venture further. Usually there is no need, they have already changed
things. Maybe the pupil just had to work at something they felt very confident with, or maybe they just liked the patterns that came from the work.

2) Sometimes when children have written something very confidently you can 'dangle a carrot' in front of them and ask them if they know anything about halves or quarters, and if so they could use them also. Very often pupils have done so when they have received no formal teaching of that subject yet.

For the exceptionally mathematically able

The pupil in this category will presumably have many more arithmetic and geometric skills and knowledge of more sophisticated processes. Then the pupil can be expected to obtain the number $8$ using their knowledge and experience.

Possible support

If a pupil has eight objects then they can access this activity by just putting the eight into a number of groups and say "this, plus this, plus this, makes 8". In this way, the eight objects can be put together and set out into different groups.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.